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QUESTION

# abstract algebra

Given:

A set S with the operation * is an Abelian group if the following five properties are shown to be true:

●  closure property: For all r and t in S, r*t is also in S

●  commutative property: For all r and t in S, r*t=t*r

●  identity property: There exists an element e in S so that for every s in S, s*e=s

●  inverse property: For every s in S, there exists an element x in S so that s*x=e

●  associative property: For every q, r, and t in S, q*(r*t)=(q*r)*t

A.  Prove that the set G (the fifth roots of unity) is an Abelian group under the operation * (complex multiplication) by using the definition given above to prove the following are true:

1.  closure property

2.  commutative property

3.  identity property

4.  inverse property

5.  associative property

Note: A Cayley table will not be sufficient to explain the associative property because this property involves three arbitrary elements, not two. To prove the associative property, express the three arbitrary elements of G using variables.

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